How can sinusoid sound be perceived as having constant loudness?

Question

To elaborate, given an audible, sinusoidal wave $A\sin(ft)$ with constant $A$ and $f$. That sounds constant to us.

Now from a few posts linked below, I understand that as stereocilia of the hair cells at the corresponding region moves, Spiral ganglion cells (SGCs) associated with these hair cells start to fire trains of neural signals with firing rate sinusoidally deviates from silent baseline rate.

Till this point, instead of something similar to power spectrograms, we just more or less have a fake "frequency axis", along with neural signal trains that are still vibration-like. I am guessing one of the following:

1. Groups of SGCs approximately cover all phases and the sum of their neural signal trains (due to rate coding) ends up with a approximately constant firing rate (and can be used as a amplitude reference)

2. Rate coding does not really go beyond giving a better vibrating train than individual SGC train. And cochlear nucleus and beyond are doing some hard work to convert the trains into something that sounds constant.

https://biology.stackexchange.com/questions/39729/depolarization-and-hyperpolarization-in-stereocilia-of-the-inner-ear

How do hair cells recognize frequencies?

How does the inner ear encode sound intensity?

How is tone volume encoded?

Answer 1

The simplest answer to the question of

How can sinusoid sound be perceived as having constant loudness?

is in the first line of your question

given an audible, sinusoidal wave $A\sin(ft)$ with constant $A$ and $f$. That sounds constant to us.

If $A$ is a constant why should the loudness vary? The answer of course is that $\sin(ft)$ is not a constant. To really answer your question, lets start by considering a different signal $A\sin(f_mt)\sin(f_ct)$ a sinusoidally amplitude modulated (SAM) tone where $f_m$ is the modulation frequency and $f_c$ is the carrier frequency. At low, but not too low, modulation rates, and carriers within the audible range, the loudness of a SAM tone varies.

To go further, we need a piece of mathematical insight. The Hilbert transform lets us decompose a signal into a slowly varying envelope and a rapidly varying fine structure. In the case of a SAM tone the envelope would be $A\sin(f_mt)$ and the fine structure would be $\sin(f_ct)$.

We have known for a long time that the auditory system is sensitive to both the temporal fine structure and the envelope. In fact, this is a key part of how cochlear implants work. While there are earlier, and more relevant, examples, Smith, Delgutte, & Oxenham (2002) is a really nice example of this duality.

The Hilbert transform is not the only way to get the envelope of a signal. You can also get the envelop by bandpass filtering, followed by half-wave rectification, followed by low pass filtering. We know the basilar membrane acts as a bandpass filter thanks to Nobel prize winning work of von Bekesky. The inner hair cells in the cochlea provide the half wave rectification. I think the field is still divided about where/how the low pass filtering happens, but there is pretty good agreement that the auditory system has a representation of the envelope.

So back to your question, to the extent that the loudness of $A\sin(ft)$ is a constant (adaptation will make the loudness decrease over time), it is because its envelope is a constant.